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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: how
Date: Mon, 22 Apr 2024 16:21:18 -0700
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On 4/22/2024 8:19 AM, WM wrote:
> Le 21/04/2024 à 18:32, Moebius a écrit :
> 
>> Alle geraden (natürlichen) Zahlen sind natürliche Zahlen. Erzählst Du 
>> Deinen Studenten an der THA etwas anderes?
> 
> No, you are right. But not all doubling results in even natural numbers.
>>
>> Proof: Let k in {2*n : n e IN}. Then there is an n in IN such that k = 
>> 2*n. Since IN is closed unter multiplication and 2 is in IN too, 2*n 
>> in IN and hence k in IN. Hence {2*n : n e IN} c IN. qed
> 
> Proof: The original set ℕ fits into (0, ω). The doubled set doesn't.

The set is infinite, not finite. Any natural number * 2 is already in 
the set. Sigh.